In this advanced probability course, we will explore several models in the mathematical theory of population genetics.

The main goal of theoretical population genetics, pioneered by the works of Fisher, Haldane, and Wright at the beginning of the 20th century, is to understand the genetic variation that can be observed in the real world. The theory successfully reconciles the Darwinian theory of evolution with Mendelian genetics.

In this course we will study several models of population genetics, under the action of different evolutionary forces. Our guiding questions, which we will tackle using the theory of Markov processes, are:
(Forward in time) What is the behaviour of the stochastic allele frequency process?
(Backward in time) What is the ancestral lineage of a subset of the current population?

Our starting point is the fundamental Wright–Fisher model, introduced to describe the phenomenon of random genetic drift. We will then consider further evolutionary forces, like selection and mutation, as well as more complicated structured models.

Keywords: Wright–Fisher model, Moran model, Kingman coalescent, Lambda coalescent, duality, Markov processes.